Introduction to Acoustics

590 Part E Music, Speech, Electroacoustics
Part E 15.2
Skeleton Curves
In this idealised model, the normal modes of the structure
at high frequencies will be similar to those of the
individual plates. Such modes will only be significantly
perturbed when the resonances of the separate plates are
close together, apart from a general background interaction
from the average weak but cumulative interaction
with all other distant modes. Woodhouse [15.76]hasrecently
shown that the averaged amplitude and phase of
the admittance can be described by skeleton curves, indicated
by the dashed lines in Fig. 15.58, on which peaks
and troughs of height Q and 1/Q and phase changes
from individual resonances are superimposed. These
curves were evaluated analytically for the rotational admittance
across the two feet of a bridge mounted on the
front plate of an idealised rectangular box-like violin
without f-holes but with a soundpost close to the treble
foot of the bridge.
At low frequencies the averaged input impedance
across the two feet of the bridge is largely reactive, with
a value close to the static springiness, which can be identified
with the low-frequency limit of the normal-mode
admittance �
n iω/mnω 2 n
, where the effective mass of
each mode will depend on where and how the instrument
is excited. However, at high frequencies the admittance
becomes largely resistive resulting from internal damping
and energy loss to the closely overlapping modes.
The use of skeleton curves enables Woodhouse to illustrate
the effect of various different bridge designs on
the overall frequency response of a violin, without having
to consider the exact positions, spacing or damping
of the individual resonances of the shell structure. Although
the idealised model is clearly over-simplistic, the
general trends predicted by such a model will clearly be
relevant to any multi-resonant shell model.
Soundpost and f-Holes
The soundpost and f-holes cut into the front plate of the
violin and related instruments have a profound effect on
the frequencies and waveforms of the normal modes,
illustrated schematically by the right-hand set of examples
in Fig. 15.57. The f-holes create an island area on
which the bridge sits, which separates the top and lower
areas of the front plate. Like the rose-hole on a guitar illustrated
in Fig. 15.56, the additional internal free edges
introduced by the f-holes tend to localise the vibrations
of the front plate to the regions above and below the island
area. In addition, the soundpost acts as a rather rigid
spring locking the vibrations of the top and back plates
together at it its ends. At low frequencies, the soundpost
introduces an approximate node of vibration on both the
top and back plates, unless the frequencies of the uncoupled
front and back plates modes are close together.
For some low-frequency modes, the soundpost and
f-hole have a relatively small effect on the modes of the
shell structure, such as the twisting mode (a) and mode
(c), when the plates vibrate in the same direction. However,
the breathing mode (c) will be strongly affected by
the soundpost forcing the front and back plates to move
together across its ends.
As indicated earlier, any string motion parallel to
the plates will exert a couple on the top of the bridge. In
the absence of the soundpost, only asymmetric modes
of the top plate could then be excited. However, to
satisfy the boundary conditions at the soundpost position,
the rocking action now induces a combination of
symmetric and antisymmetric plate modes (illustrated
schematically in Fig. 15.57b), approximately doubling
the number of modes that can contribute to the sound
of an instrument including the very important lowerfrequency
symmetric breathing modes. Because of the
f-holes, the central island can vibrate in the opposite direction
to the wings on the outer edges of the instrument.
The mixing of symmetric and antisymmetric modes is
strongly dependent on the position of the soundpost relative
to the nodes of the coupled waveforms. As a result,
the sound of a violin instrument is very sensitive to the
exact placing of the soundpost. The difference in the
sound of a violin with the soundpost first in place and
then removed is illustrated in .
To a good approximation, in the audible frequency
range, the violin soundpost can be considered as a rigid
body, as its first longitudinal resonance is ≈100 kHz,
though lower-frequency bending modes can also be excited,
particularly if the upper and lower faces of the
soundpost fail to make a flat contact with the top and
back plates (Fang and Rogers [15.95]). At high frequencies,
there is relatively little induced motion of the outer
edges of top and back plates, so that the impedance Z(ω)
(force/induced velocity) measured at the soundpost position
is simply given by the sum of the impedances at
the soundpost position, Z(ω)top + Z(ω)back, of the individual
plates with fixed outer edges. If one knows the
waveforms of the individual coupled modes, it is relatively
straightforward to evaluate the admittance at any
other point on the two surfaces, and hence to evaluate the
rotational admittance across the two feet of the bridge
(Woodhouse [15.76]).
We have already described the important role of the
bridge dynamics in the coupling between the strings and
the vibrational modes of the instrument. For instruments
of the violin family, the island region between the f-holes